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Let $\mathcal{B}$ be a homothecy invariant basis consisting of convex sets in$\mathbb{R}^n$, and define the associated geometric maximal operator$M_{\mathcal{B}}$ by $$ M_{\mathcal{B}} f(x) :=\sup_{x \in R \in\mathcal{B}}\frac{1}{|R|}\int_R |f| $$ and the halo function$\phi_{\mathcal{B}}(\alpha)$ on $(1,\infty)$ by $$\phi_{\mathcal B}(\alpha):=\sup_{E \subset \mathbb{R}^n :\, 0 < |E| < \infty}\frac{1}{|E|}|\{x\in\mathbb{R}^n : M_{\mathcal{B}} \chi_E (x) >1/\alpha\}|. $$ It is shown that if$\phi_{\mathcal{B}}(\alpha)$ satisfies the Solyanik estimate $\phi_{\mathcalB}(\alpha) - 1 \leq C (1 - \frac{1}{\alpha})^p$ for $\alpha\in(1,\infty)$sufficiently close to 1 then $\phi_{\mathcal{B}}$ lies in the H\"older class $C^p(1,\infty)$. As a consequence we obtain that the halo functions associatedwith the Hardy-Littlewood maximal operator and the strong maximal operator on$\mathbb{R}^n$ lie in the H\"older class $C^{1/n}(1,\infty)$.

Solyanik estimates and local H\"older continuity of halo functions of geometric maximal operators